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science may be generalized by the mathematicians and then be
turned back to the scientists (frequently without a solution in
any form that they can use) to be applied to new physical pro-
blems, but first and last the motive is scientific. Fourier summed
up this thesis in a famous passage which irritates one type of
mathematician, but which Poincare endorsed and followed in
much of his work.

*The profound study of nature', Fourier declared, *is the most
fecund source of mathematical discoveries. Not only does this
study, by offering a definite goal to research, have the advan-
tage of excluding vague questions and futile calculations, but
it is also a sure means of moulding analysis itself and discover-
ing those elements in it which it is essential to know and which
science ought always to conserve. These fundamental elements
are those which recur in all natural phenomena,' To which some
might retort: No doubt, but what about arithmetic hi the sense
of Gauss? However, Poincare followed Fourier's advice
whether he believed in it or not - even his researches in the
theory of numbers were more or less remotely inspired by others
closer to the mathematics of physical science.

The investigations on differential equations led out in 1880,
when Poincare was twenty-six, to one of his most brilliant dis-
coveries, a generalization of the elliptic functions (and of some
others). The nature of a (uniform) periodic function of a single
variable has frequently been described in preceding chapters,
but to bring out what Poincare did, we may repeat the essen-
tials. The trigonometric function sin z has the period 2-n-, namely,
sin (z + 2:r) = sin z\ that is, when the variable z is increased by
STT, the sine function of z returns to its initial value. For an
elliptic function, say E(z)9 there are two distinct periods, say pl
andp2> such that E(z + p^ = E(z), E(z + p^ = E(z). Poincare
found that periodicity is merely a special case of a more general
property: the value of certain functions is restored when the
variable is replaced by any one of a denumerable mfinity of
linear fractional transformations of itself., and all these trans-
formations form a group. A few symbols will clarify this state-